Examples of dense $G_\delta$ sets in $L^p$ spaces

Question

The Baire Category Theorem states that the countable intersections of open dense subsets of a complete metric space (called dense $G_\delta$ sets) are dense. Any open set is $G_\delta$, so any dense open set is dense $G_\delta$.

I know that the set of irrational numbers is an example of a dense $G_\delta$ set in $\mathbb{R}$ that is not open.

I am wondering if there are examples of non-open dense $G_\delta$ sets in function spaces such as $L^p(\mathbb{R}^n)$, $\mathcal{C}^k(\mathbb{R}^n)$, and $W^{k,p}(\mathbb{R}^n)$ endowed with their usual norms.

See Applications of the Baire category theorem by Jones (1997-1998) AND Your favourite application of the Baire Category Theorem AND Classic applications of Baire category theorem. What follows are some randomly selected references that you can use to expand your search. For $L^p$ spaces (continued) — Dave L. Renfro, Mar 06 '23 at 16:14
see Quade 1935 AND Oxtoby 1937 AND Rother 1993 AND Wingren 1999. For the space of bounded derivatives see this MSE answer. For differentiability properties of continuous functions see this MSE answer. (continued) — Dave L. Renfro, Mar 06 '23 at 16:15
For nowhere analytic $C^{\infty}$ functions see section IV here and section IV here. For topological knot theory see Milnor 1964. For topological dynamics see Akin/Hurley/Kennedy 2003. (continued) — Dave L. Renfro, Mar 06 '23 at 16:15
For mathematical logic see Suzuki 1970 AND Myhill 1960. For ODEs and PDEs see Wehausen 1945 AND Alexiewicz/Orlicz 1955-1956 AND Lasota/Yorke 1973. (continued) — Dave L. Renfro, Mar 06 '23 at 16:15
For "Riemann-rearrangemnt" series results see here. For trigonometric series convergence results see Kahane 2000. For harmonic analysis see Körner 2009. For some topological dimension results see Kuratowski 1980. For ergodic theory see Choksi/Prasad 1983. — Dave L. Renfro, Mar 06 '23 at 16:15
Basically, just google (besides google, use Zbl archive or Math. Reviews -- if you have access -- or google scholar or google books) things like "generic" or "typical" or "Baire category" along with the topic/space you are interested in, and you should come up with something. — Dave L. Renfro, Mar 06 '23 at 16:22
Here's a conference publication by MSE's Robert Israel that involves Baire category in an area not known for its use -- Some Generic Results in Mathematical Physics (2004). — Dave L. Renfro, Mar 06 '23 at 16:37

Examples of dense $G_\delta$ sets in $L^p$ spaces

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